Binomial Formulas and Factorization¶
The Three Binomial Formulas¶
First Binomial Formula¶
\[
(a + b)^2 = a^2 + 2ab + b^2
\]
Example. \((x + 3)^2 = x^2 + 6x + 9\).
Second Binomial Formula¶
\[
(a - b)^2 = a^2 - 2ab + b^2
\]
Example. \((x - 5)^2 = x^2 - 10x + 25\).
Third Binomial Formula¶
\[
(a + b)(a - b) = a^2 - b^2
\]
Example. \((x + 4)(x - 4) = x^2 - 16\).
Factorization¶
The binomial formulas can also be read in reverse — for factoring (decomposing into factors):
| Expression | Factored Form |
|---|---|
| \(a^2 + 2ab + b^2\) | \((a + b)^2\) |
| \(a^2 - 2ab + b^2\) | \((a - b)^2\) |
| \(a^2 - b^2\) | \((a + b)(a - b)\) |
Example. \(x^2 - 9 = x^2 - 3^2 = (x + 3)(x - 3)\).
Example. \(4x^2 - 12x + 9 = (2x)^2 - 2 \cdot 2x \cdot 3 + 3^2 = (2x - 3)^2\).
Application: Quadratic Equations¶
The third binomial formula solves equations of the form \(a^2 = b^2\):
\[
a^2 = b^2 \iff a^2 - b^2 = 0 \iff (a+b)(a-b) = 0 \iff a = b \text{ or } a = -b
\]
Higher Powers (Preview)¶
For \((a + b)^n\) with \(n > 2\), the binomial theorem applies:
\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]
with the binomial coefficient \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).
Example. \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\).
Summary¶
| Formula | Equation |
|---|---|
| 1st binomial | \((a+b)^2 = a^2 + 2ab + b^2\) |
| 2nd binomial | \((a-b)^2 = a^2 - 2ab + b^2\) |
| 3rd binomial | \((a+b)(a-b) = a^2 - b^2\) |
| Binomial theorem | \((a+b)^n = \sum \binom{n}{k} a^{n-k} b^k\) |
References¶
- Courant, Richard; Robbins, Herbert: What Is Mathematics? Oxford University Press, 2nd edition, 1996.